$\mathrm{PGL}_n(\mathbb{C})$-character stacks and Langlands duality over finite fields
Emmanuel Letellier, Tommaso Scognamiglio
TL;DR
This work studies the mixed Poincaré polynomials of generic $\mathrm{PGL}_n$-character stacks with local-system coefficients, proposing a combinatorial formula that is proven under Euler specialization and showing that these invariants interpolate the structure constants of dual finite-field rings of $\mathrm{PGL}_n$ and $\mathrm{SL}_n$. The approach blends intersection cohomology, weight filtrations, and twisted IC-polynomials with symmetric-function techniques to connect to Langlands duality over finite fields via convolution of IC- and character-sheaf-based bases. A central theme is the non-abelian Fourier-analytic perspective, where geometric induction and Lusztig’s character-sheaves realize dual descriptions that align under a Langlands-type correspondence between split local systems and split character-sheaves. The results illuminate a deep link between arithmetic point counts, weight-graded cohomology, and representation-theoretic multiplicities, with extensions to non-generic settings and other reductive groups discussed. Overall, the paper provides a conceptual and computational bridge between the topology of PGL$_n$-character stacks, Langlands duality over finite fields, and the algebraic structure of related based rings.
Abstract
In this paper we study the mixed Poincaré polynomial of generic $\mathrm{PGL}_n(\mathbb{C})$-character stacks with coefficients in some local systems arising from the conjugacy classes of $\mathrm{PGL}_n(\mathbb{C})$ which have non-connected stabiliser. We give a conjectural formula that we prove to be true under the Euler specialisation. We then prove that this conjectured formula interpolates the structure coefficients of the two based rings$ \left(\mathcal{C}(\mathrm{PGL}_n(\mathbb{F}_q)),Loc(\mathrm{PGL}_n),*\right)$ and $\left(\mathcal{C}(\mathrm{SL}_n(\mathbb{F}_q)), CS(\mathrm{SL}_n),\cdot\right) $ where for a group $H$, $\mathcal{C}(H)$ denotes the space of complex valued class functions on $H$, $Loc(\mathrm{PGL}_n)$ denotes the basis of characteristic functions of intermediate extensions of equivariant local systems on conjugacy classes of $\mathrm{PGL}_n$ and $CS(\mathrm{SL}_n)$ the basis of characteristic functions of Lusztig's character-sheaves on $\mathrm{SL}_n$. Our result reminds us of a non-abelian Fourier transform.